On the image of higher signature maps

TL;DR

This paper investigates the quadratic real cycle class map from Chow-Witt groups to cohomology groups of real algebraic varieties, focusing on specific codimensions and proposing a conjecture on its image.

Contribution

It formulates a precise conjecture describing the image of the real cycle class map in certain codimensions, supported by new results.

Findings

Conjecture on the image of the cycle class map in codimensions 0, d-2, d-1, d.

Results corroborate the conjecture in these specific cases.

Provides a detailed analysis of the cokernel exponents of the map.

Abstract

Given a smooth variety $X$ over the field $\mathbb{R}$ of real numbers and a line bundle $\mathcal{L}$ on $X$ with associated topological line bundle $L=\mathcal{L}(\mathbb{R})$, we study the quadratic real cycle class map $\widetilde{\gamma}_{\mathbb{R}}^c:\widetilde{\mathrm{CH}}^c(X,\mathcal{L})\rightarrow\mathrm{H}^c(X(\mathbb{R}),\mathbb{Z}(L))$ from the $c$-th Chow-Witt group of $X$ to the $c$-th cohomology group of its real locus $X(\mathbb{R})$ with coefficients in the local system $\mathbb{Z}(L)$ associated with $L$. We focus on the cases $c\in\{0,d-2,d-1,d\}$ where $d$ is the dimension of $X$ and we formulate a precise conjecture on the image of $\widetilde{\gamma}_{\mathbb{R}}$ in terms of the exponents of its cokernel that is corroborated by the results obtained in those codimensions.